1. Field of the Invention
This invention relates broadly to methods and apparatus for investigating subsurface earth formations. More particularly, this invention relates to sonic borehole tools and methods for measuring a nonlinear parameter of an earth formation. The invention has particular application in using the nonlinear parameter measurements for determining in situ the strength of rocks, which information is useful in the production of oil from the formation, although the invention is not limited thereto and provides other useful information regarding formation parameters.
2. State of the Art
The art of sonic well logging for use in determining formation parameters is a well established art. Sonic well logs are typically derived from sonic tools suspended in a mud-filled borehole by a cable. The tools typically include a sonic source (transmitter) and a plurality of receivers which are spaced apart by several inches or feet. Typically, a sonic signal is transmitted from the transmitter at one longitudinal end of the tool and received by the receivers at the other, and measurements are made every few inches as the tool is drawn up the borehole. Depending upon the type of transmitter or source utilized (e.g., dipole, monopole), the sonic signal generated by the transmitter travels up the borehole and/or enters the formation adjacent the borehole, and the arrival times of one or more of the compressional (P-wave), shear (S-wave), Stoneley (tube wave), and flexural wave can be detected by the receivers. The receiver responses are typically processed in order to provide a time to depth conversion capability for seismic studies as well as for providing the determinations of formations parameters such as porosity.
A method of using a sonic tool in conjunction with the changing of the pressure in the borehole is also known. In the defensive publication H1156 of Siegfried, it is suggested that compressional and shear wave speeds, amplitudes, and phase shifts of sonic waveforms be measured in the borehole during multiple runs by a sonic logging tool, where the borehole pressure is different for each of the runs. According to Siegfried, differences in any of these acoustic properties resulting from a change in pressure provides an indication of the relative fracturing of the formation.
While measurements of the compressional and shear waves are useful in quantifying and characterizing various parameters of the formation, including fracturing, it will be appreciated that to date, there has been no successful mechanism for making in situ determinations of nonlinear aspects of the formation. For purposes of this invention, it should be understood that the term "nonlinear" when used to describe a material relates to the fact that a plot of stress versus strain in a material will exhibit some nonlinear behavior. Phenomenologically, the strain energy function U(e) of an isotropic elastic solid can be written as: EQU U(.epsilon.)=f(.lambda.,.mu.).epsilon..sup.2 +g(.alpha., .beta., .gamma.).epsilon..sup.3 ( 1)
where .epsilon. is the strain, .lambda. and .mu. are the second order elastic Lame constants, and .alpha., .beta., and .gamma. are the third order elastic constants. From equation (1), it will be appreciated that the stress a may be defined by: EQU .sigma.=U/ .epsilon.)=f (.lambda.,.mu.).epsilon.+g(.alpha.,.beta.,.gamma.).epsilon..sup.2 ( 2)
where f and g denote general functions of quantities in parentheses. Based on equation (2) it is seen that the second order Lame constants are linear constants, while the third order constants are nonlinear, and hence measure the nonlinearity of the material. The more nonlinear the stress versus strain plot is, the more nonlinear the material is said to be. Various manifestations of non-linearity include: the varying of the acoustic velocity in the material when the confining pressure changes; the varying of the acoustic velocity in the material when the amplitude of the acoustic wave changes; the interaction of two monochromatic acoustic beams having different frequencies to create third and fourth acoustic beams having the difference frequency and the additive frequency of the two incident beams; and evidence of frequencies being generated within the material which were not part of any input signal.
In the oil production industry, rock properties such as sanding, fracturing and borehole collapse can be considered to relate to the nonlinear properties of the formation. In each case, the strain in the rock catastrophically exceeds that which would be expected from a linear stress-strain relationship. As suggested in the parent applications hereto, since the less consolidated a formation is, the more nonlinear it is, a measurement of the nonlinearity of the formation can provide a measurement of the relative state of the consolidation of the formation. As suggested above, whether a layer of a formation is well or poorly consolidated, can broadly affect the producibility of the layer and formation, as well as the manner in which production is to be carried out.